Motivated by Caflish's work [1]} and the improvement work Guo-Jang-Jiang [2]}, in the present paper we expect to employ Hilbert expansion approach to rigorously justify the higher order hydrodynamic corrections to the compressible Euler system from (REF )-(REF ), as \(\varepsilon \rightarrow 0\) . However, the more terms are expanded, the more special the solutions are. In this sense, the ansatz of truncating Hilbert expansion is taking the following form
\(F_\varepsilon =\mu +\sum _{n=1}^{3}\varepsilon ^n F_n+\varepsilon ^3 F_{R,\varepsilon },\) 
